Multiple vaccines may make them less effective.

Source: Massimo Fabiani, Enrico Volpe, Maurizio Faraone, Antonino Bella, Patrizio Pezzotti, Francesco Chini. (2020) Effectiveness of influenza vaccine in reducing influenza-associated hospitalizations and deaths among the elderly population; Lazio region, Italy, season 2016-2017. Expert Review of Vaccines 19:5, pages 479-489.

A meta-analysis of vaccine effectiveness studies from 2010–11 through 2014–15 shows substantial heterogeneity in repeat vaccination effects within and between seasons and subtypes. When negative effects were observed, they were most pronounced for H3N2, especially in 2014–15 when vaccine components were unchanged and antigenically distinct from the epidemic strain.

Thus in some circumstances, multiple vaccines may render them less effective for the person.